22 July 2021
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Original Citation:
Higher Hamming weights for locally recoverable codes on algebraic curves
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DOI:10.1016/j.ffa.2016.03.004 Terms of use:
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H I G H E R H A M M I N G W E I G H T S F O R
L O C A L L Y R E C O V E R A B L E C O D E S
O N A L G E B R A I C C U R V E S
Edoardo Ballico
∗& Chiara Marcolla
1abstract
We study locally recoverable codes on algebraic curves. In the first part of the manuscript, we provide a bound on the generalized Hamming weight of these codes. In the second part, we propose a new family of algebraic geometric LRC codes, which are LRC codes from the Norm-Trace curve. Finally, using some properties of Hermitian codes, we improve the bounds on the distance proposed in [1] of some Hermitian LRC codes.
1
introduction
The v-th generalized Hamming weight dv(C) of a linear code C is the minimum
sup-port size of v-dimensional subcodes of C. The sequence d1(C), . . . , dk(C)of generalized
Hamming weights was introduced by Wei [37] to characterize the performance of a
lin-ear code on the wire-tap channel of type II. Later, the GHWs of linlin-ear codes have been used in many other applications regarding the communications, as for bounding the covering radius of linear codes [15], in network coding [26], in the context of list
decod-ing [7, 9], and finally for secure secret sharing [18]. Moreover, in [2] the authors show
in which way an arbitrary linear code gives rise to a secret sharing scheme, in [16, 17]
the connection between the trellis or state complexity of a code and its GHWs is found and in [4] the author proves the equivalence to the dimension/length profile of a code
and its generalized Hamming weight. For these reasons, the GHWs (and their extended version, the relative generalized Hamming weights [21,19]) play a central role in coding
theory. In particular, generalized and relative generalized Hamming weights are studied for Reed-Muller codes [10,23] and for codes constructed by using an algebraic curve [6] ∗ The first author is partially supported by MIUR and GNSAGA of INDAM (Italy).
* Department of Mathematics, University of Trento, Italy. Email address: ballico@science.unitn.it 1
Department of Mathematics, University of Turin, Italy. Email address: chiara.marcolla@unito.it
preliminary notions 2
as Goppa codes [24,38], Hermitian codes [12,25] and Castle codes [27].
In this paper, we provide a bound on the generalized Hamming weight of locally re-coverable codes on the algebraic curves proposed in [1]. Moreover, we introduce a new
family of algebraic geometric LRC codes and improve the bounds on the distance for some Hermitian LRC codes.
Locally recoverable codes were introduced in [8] and they have been significantly
studied because of their applications in distributed and cloud storage systems [3,13,32, 34,35]. We recall that a code C ∈ (Fq)n has locality r if every symbol of a codeword c
can be recovered from a subset of r other symbols of c.
In other words, we consider a finite field K = Fq, where q is a power of a prime, and
an [n, k] code C over the field K, where k = logq(|C|). For each i ∈ {1, . . . , n} and each
a∈ K set C(i, a) ={c ∈ C | ci = a}. For each I ⊆ {1, . . . , n} and each S ⊆ C let SI be the
restriction of S to the coordinates in I.
Definition 1. Let C be an [n, k] code over the field K, where k = logq(|C|). Then C is
said to have all-symbol locality r if for each a ∈ Fq and each i ∈ {1, . . . , n} there is
Ii⊂{1, . . . , n} \ {i} with |Ii| 6 r, such that for CIi(i, a) ∩ CIi(i, a
0) =∅ for all a 6= a0. We
use the notation (n, k, r) to refer to the parameters of this code.
Note that if we receive a codeword c correct except for an erasure at i, we can recover the codeword by looking at its coordinates in Ii. For this reason, Iiis called a recovering
set for the symbol ci.
Let C be an (n, k, r) code, then the distance of this code has to verify the bound proved in [28,8] that is d 6 n − k − dk/re + 2. The codes that achieve this bound with equality
are called optimal LRC codes [32,34,35]. Note that when r = k, we obtain the Singleton
bound, therefore optimal LRC codes with r = k are MDS codes.
layout of the paper This paper is divided as follows. In Section 2 we recall the
notions of algebraic geometric codes and the definition of algebraic geometric locally recoverable codes introduced in [1]. In Section3we provide a bound on the generalized
Hamming weights of the latter codes. In Section4we propose a new family of algebraic
geometric LRC codes, which are LRC codes from the Norm–Trace curve. Finally, in Section5 we improve the bounds on the distance proposed in [1] for some Hermitian
LRC codes, using some properties of the Hermitian codes.
2
preliminary notions
2.1 Algebraic geometric codes
Let K =Fqbe a finite field, where q is a power of a prime. LetX be a smooth projective
func-preliminary notions 3
tions field onX. Let D be a divisor on the curve X. We recall that the Riemann-Roch space associated to D is a vector spaceL(D) over K defined as
L(D) = {f ∈ K(X) | (f) + D > 0} ∪ {0}. where we denote by (f) the divisor of f.
Assume that P1, . . . , Pn are rational points on X and D is a divisor such that D =
P1+ . . . + Pn. Let G be some other divisor such that supp(D) ∩ supp(G) = ∅. Then we
can define the algebraic geometric code as follows:
Definition 2. The algebraic geometric code (or AG code) C(D, G) associated with the divisors D and G is defined as
C(D, G) ={(f(P1), . . . , f(Pn))| f ∈ L(G)} ⊂ Kn.
The dual C⊥(D, G) of C(D, G) is an algebraic geometric code.
In other words an algebraic geometric code is the image of the evaluation map Im(evD) =
C(D, G), where the evaluation map evD:L(G) → Knis given by
evD(f) = (f(P1), . . . , f(Pn))∈ Kn.
Note that if D = P1+ . . . + Pn and we denote byP = {P1, . . . , Pn} we can also indicate
evDas evP.
2.2 Algebraic geometric locally recoverable codes
In this section we consider the construction of algebraic geometric locally recoverable codes of [1].
LetX and Y be smooth projective absolutely irreducible curves over K. Let g : X → Y be a rational separable map of curves of degree r + 1. Since g is separable, then there exists a function x ∈ K(X) such that K(X) = K(Y)(x) and that x satisfies the equation xr+1+ brxr+ . . . + b0 = 0, where bi∈ K(Y). The function x can be considered as a map
x :X → PK. Let h = deg(x) be the degree of x.
We consider a subset S = {P1, . . . , Ps} ⊂ Y(K) of Fq-rational points of Y, a divisor Q∞
such that supp(Q∞)∩ supp(S) = ∅ and a positive divisor D = tQ∞. We denote by
A = g−1(S) ={P
ij, where i = 0, . . . , r, j = 1, . . . , s} ⊂ X(K),
where g(Pij) = Pi for all i, j and assume that bi are functions in L(niQ∞) for some
natural numbers niwith i = 1, . . . , r.
Let{f1, . . . , fm} be a basis of the Riemann-Roch space L(D). By the Riemann-Roch
The-orem we have that m > deg(D) + 1 − gY, where gYis the genus ofY.
From now on, we assume that m = deg(D) + 1 − gY, where deg(D) = t`, and we consider the K-subspace V of K(X) of dimension rm generated by
generalized hamming weights of ag lrc codes 4
We consider the evaluation map evA : V → K(r+1)s. Then we have the following
theo-rem.
Theorem 1. The linear space C(D, g) = SpanK(r+1)shevA(B)i is an (n, k, r) algebraic
geomet-ric LRC code with parameters
n = (r + 1)s
k = rm> r(t` + 1 − gY) d > n − t`(r + 1) − (r − 1)h. Proof. See Theorem 3.1 of [1].
The AG LRC codes have an additional property. They are LRC codes (n, k, r) with (r + 1)| n and r | k. The set {1, . . . , n} can be divided into n/(r + 1) disjoint subsets Uj for 1 6 j 6 s with the same cardinality r + 1. For each i the set Ii ⊆ {1, . . . , n} \ {i} is the complement of i in the element of the partition Ujcontaining j, i.e. for all i, j ∈{1, . . . , n}
either Ii= Ij or Ii∩ Ij =∅.
Moreover, they have also the following nice property. Fix w ∈ (K)n and denote by wUj ={wι, for any ι ∈ Uj}. Suppose we receive all the symbols in Uj. There is a simple
linear parity test on the r + 1 symbols of Uj such that if this parity check fails we know
that at least one of the symbols in Uj is wrong. If we are guaranteed (or we assume)
that at most one of the symbols in Uj is wrong and the parity check is OK, then all the
symbols in Uj are correct. Moreover we can recover an erased symbol wι, with ι ∈ Uj
using a polynomial interpolation through the points of the recovering set wUj.
3
generalized hamming weights of ag lrc codes
Let K be a field and letX be a smooth and geometrically connected curve of genus g > 2 defined over the field K. We also assumeX(K) 6= ∅. We recall the following definitions:
Definition 3([29], [30]). The K-gonality γK(X) of X over a field K is the smallest possible
degree of a dominant rational mapX → P1
K. For any field extension L of K, we define
also the L-gonality γL(X) of X as the gonality of the base extension XL = X ×KL. It is
an invariant of the function field L(X) of XL.
Moreover, for each integer i > 0, the i-th gonality γi,L(X) of X is the minimal degree z
such that there is R ∈ Picz(X)(L) with h0(R)> i + 1. The sequence γ
i,K(X) is the usual
gonality sequence [20]. Moreover, the integer γ1,K(X) = γK(X) is the K-gonality of X.
Let K = Fq a finite field with q elements. Let C ⊂ Kn be a linear [n, k] code over K.
We recall that the support of C is defined as follows
supp(C) ={i | ci6= 0 for some c ∈ C}.
So ]supp(C) is the number of nonzero columns in a generator matrix for C. Moreover, for any 1 6 v 6 k, the v-th generalized Hamming weight of C [14, §7.10], [36, §1.1] is
defined by
lrc codes from norm–trace curve 5
In other words, for any integer 1 6 v 6 k, dv(C) is the v-th minimum support weights,
i.e. the minimal integer t such that there are an [n, v] subcode D of C and a subset S⊂ {1, . . . , n} such that ](S) = t and each codeword of D has zero coordinates outside S. The sequence d1(C), . . . , dk(C) of generalized Hamming weights (also called weight
hierarchy of C) is strictly increasing (see Theorem 7.10.1 of [14]). Note that d1(C) is the
minimum distance of the code C.
Let us consider X and Y smooth projective absolutely irreducible curves over K and let g : X → Y be a rational separable map of curves of degree r + 1. Moreover we take r, t, Q∞, f1, . . . , fm andA = g−1(S)defined as Section 2.2. So we can construct an
(n, k, r) algebraic geometric LRC code C as in Theorem 1. For this code we have the
following:
Theorem 2. Let C be an (n, k, r) algebraic geometric LRC code as in Theorem 1. For every integer v > 2 we have that
dv(C)> n − t`(r + 1) − (r − 1)h + γv−1,K(X).
Proof. Take a v-dimensional linear subspaceD of C and call E⊆{Pij| i = 0, . . . r, j = 1, . . . , s},
the set of common zeros of all elements of D. Since n − dv(C) = ](E), we have to
prove that t`(r + 1) + (r − 1)h − ](E) > γv−1,K(X). Fix u ∈ D \ {0} and let Fu denote
the zeros of u. Note that Fu is contained in the set {Pij | i = 0, . . . r, j = 1, . . . , s} by
the definition of the code C. We have Fu ⊇ E. By the definition of the integers t, ` and
h :=deg(x), we have ](Fu)6 t`(r + 1) + (r − 1)h. The divisors Fu− E, u ∈D \ {0} form a
family of linearly equivalent non-negative divisors, each of them defined over K. Since dim(D) = v, the definition of γv−1,K(X) gives ](Fu) −](E) > γv−1,K(X). This inequality for a single u ∈D \ {0} proves the theorem.
See Remark1for an application of Theorem2.
4
lrc codes from norm–trace curve
In this section we propose a new family of Algebraic Geometric LRC codes, that is, a LRC codes from the Norm–Trace curve. Moreover, we compute theFqu-gonality of the
Norm-Trace curve.
Let K = Fqu be a finite field, where q is a power of a prime. We consider the norm
NFquFq and the trace TrFquFq , two functions from Fqu toFqdefined as
NFquFq (x) = x1+q+···+qu−1 and TrFquFq (x) = x + xq+· · · + xqu−1.
The Norm-Trace curve χ is the curve defined over K by the following affine equation NFquFq (x) =TrFquFq (y),
lrc codes from norm–trace curve 6
that is,
x(qu−1)/(q−1)= yqu−1+ yqu−2+ . . . + ywhere x, y ∈ K (1) The Norm-Trace curve χ has exactly n = q2u−1K-rational affine points (see Appendix A
of [5]), that we denote byPχ={P1, . . . , Pn}. The genus of χ is g = 1
2(q
u−1− 1)(qu−1
q−1 − 1).
Note that if we consider u = 2, we obtain the Hermitian curve.
Starting from the Norm–Trace curve, we have two different ways to construct Norm– Trace LRC codes.
projection on x We have to construct a qu-ary (n, k, r) LRC codes. We consider
the natural projection g(x, y) = x. Then the degree of g is qu−1 = r + 1 and the
degree of y is h = 1 + q + · · · + qu−1.
To construct the codes we consider S = Fqu and D = tQ∞for some t > 1. Then, using
a construction of Theorem1we find the parameters for these Norm–Trace LRC codes.
Proposition 1. A family of Norm–Trace LRC codes has the following parameters:
n = q2u−1, k = mr = (t + 1)(qu−1− 1) and
d> n − tqu−1 − (qu−1 − 1)(1 + q +· · · + qu−1).
projection on y We have to construct a qu-ary (n, k, r) LRC codes. We consider
the other natural projection g0(x, y) = y. Then deg(g0) = 1 + q +· · · + qu−1 = r + 1.
In this case we take S =Fqu\M, where
M = {a ∈ Fqu | aq u−1
+ aqu−2+ . . . + a = 0},
so r = q + · · · + qu−1 and h = deg(x) = qu−1. Then, using Theorem1we have the
following
Proposition 2. A family of Norm–Trace LRC codes has the following parameters:
n = q2u−1− qu−1, k = mr = (t + 1)(q +· · · + qu−1)
and
d> n − tqu−1− (q +· · · + qu−1) − qu−1(qu−1+· · · + q − 1).
For the Norm–Trace curve χ we are able to find the K-gonality of χ.
Lemma 1. Let χ be a Norm–Trace curve defined overFqu, where u > 2. We have γ1,Fqu(χ) =
qu−1.
Proof. The linear projection onto the x axis has degree qu−1 and it is defined over Fq
and hence over Fqu. Thus γ1,Fqu(χ) 6 qu−1. Denote by z = γ1,Fqu(χ) and assume
that z 6 qu−1− 1. By the definition of K-gonality, there is a non-constant morphism
w : χ→P1with deg(w) = z and defined overF
qu. Since w(χ(Fqu))⊆P1(Fqu), we get
hermitian lrc codes 7
Remark 1. By Lemma1, we can apply Theorem 2to the Norm–Trace curve. In fact, we
can consider the gonality sequence over K of χ to get a lower bound on the second generalized Hamming weight of the two families of Norm–Trace LRC codes:
• Let t > 1 and let C be a (q2u−1, (t + 1)(qu−1− 1), qu−1− 1) Norm–Trace LRC
code. Then we have
d2(C)> q2u−1+ qu−1− tqu−1− (qu−1− 1)(1 + q +· · · + qu−1).
• Let t > 1 and let C be a Norm–Trace LRC code with parameters (q2u−1− qu−1,
(t + 1)(q +· · · + qu−1), q + · · · + qu−1). Then we have
d2(C)> q2u−1− (t − 1)qu−1− (1 + qu−1)(q +· · · + qu−1).
5
hermitian lrc codes
In this section we improve the bound on the distance of Hermitian LRC codes proposed in [1] using some properties of Hermitian codes which are a special case of algebraic
geometric codes.
5.1 Hermitian codes
Let us consider K =Fq2 a finite field with q2 elements. The Hermitian curveH is defined
over K by the affine equation
xq+1= yq+ y where x, y ∈ K. (2) This curve has genus g = q(q−1)2 and has q3+ 1 points of degree one, namely a pole Q∞ and n = q3 rational affine points, denoted byPH={P1, . . . , Pn} [31].
Definition 4. Let m ∈N such that 0 6 m 6 q3+ q2− q − 2. Then the Hermitian code
C(m, q) is the code C(D, mQ∞)where
D = X
αq+1=βq+β
Pα,β
is the sum of all places of degree one (except Q∞, that is a point at infinity) of the Hermitian function field K(H).
By Lemma 6.4.4. of [33] we have that
Bm,q ={xiyj | qi + (q + 1)j 6 m, 0 6 i 6 q2− 1, 0 6 j 6 q − 1},
forms a basis ofL(mQ∞). For this reason, the Hermitian code C(m, q) could be seen as SpanFq2hevPH(Bm,q)i. Moreover, the dual of C(m, q) denoted by C(m⊥, q) = C⊥(m, q)
is again an Hermitian code and it is well known (Proposition 8.3.2 of [33]) that the
degree m of the divisor has the following relation with respect to m⊥:
hermitian lrc codes 8
The Hermitian codes can be divided in four phases [11], any of them having specific
explicit formulas linking their dimension and their distance [22]. In particular we are
interested in the first and the last phase of Hermitian codes, which are:
i phase: 0 6 m⊥ 6 q2− 2 . Then we have m⊥ = aq + b where 0 6 b 6 a 6 q − 1
and b 6= q − 1. In this case, the distance is
d = a + 1 if a > b
d = a + 2 if a = b. (4)
iv phase: n − 1 6 m⊥ 6 n + 2g − 2 . In this case m⊥ = n + 2g − 2 − aq − bwhere
a, b are integers such that 0 6 b 6 a 6 q − 2 and the distance is
d = n − aq − b. (5)
5.2 Bound on distance of Hermitian LRC codes
Let K = Fq2 be a finite field, where q is a power of a prime. Let X = H be the
Hermitian curve with affine equation as in (2). We recall that this curve has q3 F
q2
-rational affine points plus one at infinity, that we denoted by Q∞.
We consider two of the three constructions of Hermitian LRC codes proposed in [1]
and we improve the bound on distance of Hermitian LRC codes using properties of Hermitian codes. In particular, if we find an Hermitian code C(m, q) = CHer such
that CLRC ⊂ CHer, then we have dLRC > dHer.
projection on x By Proposition 4 of [1], we have a family of (n, k, r) Hermitian
LRC codes with r = q − 1, length n = q3, dimension k = (t − 1)(q − 1) and distance
d > n − tq − (q − 2)(q + 1). Moreover, for these codes, S = K, D = tQ∞ for some 1 6 t 6 q2 − 1and the basis for the vector space V is
B = {xjyi | j = 0, . . . , t, i = 0, . . . , q − 2}.
(6) Using the Hermitian codes, we improve the bound on the distance for any integer t, such that q2− q + 1 6 t 6 q2 − 1.
To find an Hermitian code C(m, q) = CHer such that CLRC ⊂ CHer, we have to
compute the setBm,q, that is, we have to find m. After that, to compute the distance of
C(m, q) we use (4) and (5).
We consider the first Hermitian phase: 0 6 m⊥ 6 q2 − 2, that is, q2 − q + 1 6 t 6
q2− 1.
For this phase m⊥ = aq + b, where 0 6 b 6 a 6 q − 1 and the distance of the
Hermitian code is either d = a + 1 if a > b or d = a + 2 if a = b. By (6), m must be
equal to m = qt + (q + 1)(q − 2) and by (3) we have that m⊥ = n + 2g − 2 − m =
q(q2 − t). So b = 0 and a = q2 − t and the distance of the Hermitian code is dHer = a + 1 = q2 − t + 1, since a > b. This implies that
references 9
Note that (7) improves the bound on the distance proposed in Proposition 4 of [1] since q2− t + 1 > q3− tq − (q − 2)(q + 1) ⇐⇒ t(q − 1) > q(q − 1)2+ 1 ⇐⇒ t > q2− q.
We just proved the following:
Proposition 3. Let q2− q + 1 6 t 6 q2− 1. It is possible to construct a family of (n, k, r) Hermitian LRC codes{Ct}q2−q+16t6q2−1 with the following parameters:
n = q3, k = (t − 1)(q − 1), r = q − 1 and d > q2− t + 1.
two recovering sets In [1] the authors propose an Hermitian code with two
recov-ering sets of size r1= q − 1and r2 = q, denoted by LRC(2). They consider
L = Span{xiyj, i = 0, . . . , q − 2, j = 0, . . . , q − 1}
and a linear code C obtained by evaluating the functions in L at the points of B = g−1(Fq2\M), where g(x, y) = x and M = {a ∈ Fq | aq+ a = 0}. So |B| = q3− q. By
Proposition 4.3 of [1], the LRC(2) code has length n = (q2− 1)q, dimension k = (q − 1)q
and distance
d> (q + 1)(q2− 3q + 3) = q3− 2q2+ 3. (8) As before, we improve the bound on the distance using Hermitian codes that contains the LRC(2) code. To do this we have to find m⊥. By L, we have that m = q(q −
1) + (q + 1)(q − 2) so we are in the fourth phase of Hermitian codes because m⊥ =
n + 2g − 2 − m = q3− q2+ q. In this case dHer = m⊥− 2g + 2 = q3+ 2q + 2. Since
|B| = q3− q, we have that
dLRC> dHer− q = q3+ q + 2. (9) Note that this bound improves bound (8). We just proved the following proposition:
Proposition 4. Let C be a linear code obtained by evaluating the functions in L at the points of
B. Then C has the following parameters:
n = (q2− 1)q, k = (q − 1)q, r1= q − 1, r2 = qand d > q3+ q + 2.
acknowledgements
The authors would like to thank the anonymous referees for their comments.
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